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The purpose of this work is to present the atomic model proposed by Sommerfeld. We outline the classical calculation of the elliptical orbit and then apply the Wilson–Sommerfeld quantization rules to obtain expressions for the quantization of energy and orbit semi-axes. We then apply the relativistic theory to solve the problem of degenerate orbits. Thus, we see that the Sommerfeld atom is a very rich topic, involving the integration of different subjects ranging from Lagrangian mechanics to relativity, as well as plane geometry and differential equations. Editor's Note: The authors present Sommerfeld s atomic model as a valuable teaching tool, both because it incorporates many important topics in physics instruction and because it helps students appreciate the development of scientific ideas. Readers will appreciate the careful presentation of the physics. [ABSTRACT FROM AUTHOR]

The given text discusses different approaches to finding equations of motion for physical systems, such as the Newtonian, Lagrangian, and Hamiltonian approaches. It presents a supplementary tool using the total differential of the mechanical energy function to obtain equations of motion for certain systems. The text also explores the relationship between this approach and Hamilton's equations of motion. Additionally, the text provides concise and factual information about different types of motion and their corresponding equations. [Extracted from the article]

In this study, we analyze a symmetrical anharmonic oscillator that differs from a basic harmonic oscillator by including additional terms in the potential energy function. This oscillator's nonlinear characteristics are important in many physics fields, allowing for modeling of complex systems. We begin by creating the Lagrangian and obtaining the equation of motion through the Euler–Lagrange equation. Both the multi-step differential transforms method (Ms-DTM) and the Runge–Kutta 4th-order (RK4) method are employed to solve this equation, providing both analytical and numerical solutions. By examining these solutions, we confirm our findings and enhance our comprehension of the oscillator's behavior, which can be applied to more intricate nonlinear systems. [ABSTRACT FROM AUTHOR]

In this investigation, a magnetically coupled double-spring design is presented for harvesting low-level non-stationary random vibrational energy. The sensor relies on multimodal coupling between the translation and rotation of a two-spring magnet and coil system to widen the harvesting bandwidth. Energy methods are used to develop a model to characterize the electromechanical response of the system, the solution of which is obtained using stochastic techniques based on a particle swarm algorithm. This approach provides an efficient method to estimate system parameters that otherwise are difficult or impossible to determine with independent measurements. The experimental results demonstrate agreement with the theoretical predictions over a limited bandwidth. The sensor can effectively harvest non-stationary vibration energy down to 10−4 g within a limited bandwidth of 130–150 Hz. The sensor prototype has an operational volume of 2.6 c m 3 with a calculated power density of 0.2 W / c m 3 . The sensor's small size results in a coupling efficiency of approximately 6% across the tested bandwidth. [ABSTRACT FROM AUTHOR]

Quantum optimal control (QOC) is a set of methods for designing time-varying electromagnetic fields to perform operations in quantum technologies. This tutorial paper introduces the basic elements of this theory based on the Pontryagin maximum principle, in a physicist-friendly way. An analogy with classical Lagrangian and Hamiltonian mechanics is proposed to present the main results used in this field. Emphasis is placed on the different numerical algorithms to solve a QOC problem. Several examples ranging from the control of two-level quantum systems to that of Bose–Einstein condensates (BECs) in a one-dimensional optical lattice are studied in detail, using both analytical and numerical methods. Codes based on shooting method and gradient-based algorithms are provided. The connection between optimal processes and the quantum speed limit is also discussed in two-level quantum systems. In the case of BEC, the experimental implementation of optimal control protocols is described, both for two-level and many-level cases, with the current constraints and limitations of such platforms. This presentation is illustrated by the corresponding experimental results. [ABSTRACT FROM AUTHOR]

The finite amplitude radial motion of thick walled hyperelastic tubes has been extensively studied since the work of Knowles in the context of material incompressibility. This allows for explicit integration of the balance equation for linear momentum in the radial direction. We use this procedure to examine the effect of layering upon the oscillatory response of such tubes. For a suite of materials characterized by different hyperelastic constitutive laws and different material densities, we consider the parametric effect of property mismatch, relative layer thickness, and layer ordering on the qualitative shape of closed orbits in a phase space of radial displacement vs. radial velocity. Even the simple case of a two-layer system shows how changing a single parameter can give significant qualitative variation in orbital shape (e.g., eccentricity, crowding of maximum velocity locations toward minimum displacement locations, etc.) which in turn alters the orbital period (oscillation frequency). Changes in more than one parameter can either exacerbate or reduce such effects, in the latter case by for example a balancing of shear wave speeds. Equivalent results are obtained by a somewhat more direct energy treatment, either using Lagrangian dynamics or Hamilton's principle, both of which bypass the notion of stress. [ABSTRACT FROM AUTHOR]

Robot is an indispensable technology within the context of the Industrial 4.0 revolution, providing a wide range of applications in industry. The stability of a robot's manipulator is contingent upon its associated manipulator settings, hence influencing its overall quality. The assessment of stability has been examined in prior studies using a limited set of manipulator parameters, leading to a deficient comprehension of this phenomenon. In this study, a mathematical model of a flexible six-link manipulator is formulated using Lagrangian mechanics and various parameters. An innovative mathematical framework is developed to establish a correlation between stability and the motor acceleration and moment of inertia for the Universal Robot (UR5). Furthermore, extensive research has been conducted to examine the correlation between stiffness, damping, and deflection in the context of stability. In spite of this, fuzzy logic inference methods are employed to ascertain the relative significance of stiffness, damping, and deflection with regards to stability. Mathematical approaches are employed to validate the numerical values of several manipulator parameters. The findings of the study demonstrate that there is a positive correlation between motor acceleration and stability, indicating that as motor acceleration grows, stability also increases. Conversely, there is a negative correlation between stability and moment of inertia, suggesting that stability reduces as moment of inertia increases. In the case of stiffness, damping, and deflection, stiffness is the most important factor. When stiffness is high, stability is also high. However, when stiffness is minimal, stability is low. The implications of these findings are expected to have a positive impact on enhancing industrial productivity.

This paper establishes the concept of elastic wave propagation in a multi-porous medium with different permeabilities by assuming there are n distinct pore fluid phases. The dynamic equation of motion of elastic wave propagation through this multi-porous medium is derived based on Lagrangian mechanics. In this regard, the generalized form of mass coefficients and then the energy loss due to the fluid phases in terms of dissipation coefficients are presented for low-frequency limits with the help of Darcy's Law of multi-phases system. The elastic coefficients of the constitutive equation in terms of compliance matrix are identified using a series of Gedanken experiments. Some significant results regarding the compressional and rotational waves in a multi-porosity medium are derived. The validation of the theory has been shown by comparing it with the existing theory of single and double porosity. It is observed that there are (n+1) compressional waves corresponding to solid and fluid phases, whereas only one rotational wave is associated with the solid phase. The concept of multi-porosity theory can contribute to a deeper understanding of wave behaviour in a porous medium. [ABSTRACT FROM AUTHOR]

When foehn winds surmount the Alps from the south, they often abruptly and vigorously descend into the leeside valleys on the Alpine north side. Scientists have long been intrigued by the underlying cause of this pronounced descent. While mountain gravity waves and the hydraulic theory provide theoretical foundations to explain the phenomenon, the descent of the Alpine south foehn has, so far, not been explicitly quantified and characterized for a series of real-case events. To fill this research gap, the present study employs kilometer-scale numerical simulations, combined with online trajectories calculated during model integration. In an innovative approach, we adopt the Lagrangian perspective, enabling us to identify the descent and determine its key characteristics across foehn regions spanning from the Western to the Eastern Alps. In the first part of the study, we find the descent of foehn air parcels to be primarily confined to distinct hotspots in the immediate lee of local mountain peaks and chains, underlining the fundamental role of local topography in providing a natural anchor for the descent during south foehn. Consequently, the small-scale elevation differences in the underlying terrain are clearly linked to the magnitude of the descent, whereby other contributing factors also influence the process. Combined with the fact that the descent is mostly dry adiabatic, these results suggest that the descending motion occurs along downward-sloping isentropes associated with gravity waves. A small proportion of air parcels experience diabatic cooling and moisture uptake during the descent, which predominantly occur to the south of the Alpine crest. The second part of the study aims to elucidate the different factors affecting the descent on a local scale. To this end, a particularly prominent hotspot situated along the Rätikon, a regional mountain range adjacent to the Rhine Valley, is examined in two detailed case studies. During periods characterized by intensified descent, local peaks along the Rätikon excite gravity waves that are linked to the descent of air parcels into the northern tributaries of the Rätikon and into the Rhine Valley. The two case studies reveal that different wave regimes, including vertically propagating waves, breaking waves, and horizontally propagating lee waves, coincide with the descent. This suggests the absence of a specific wave regime that is consistently present during foehn descent periods along the Rätikon. In addition to gravity waves, other effects likewise influence the descent activity. For example, a topographic concavity deflects the near-surface flow and thus promotes strong descent of air parcels towards the floor of the Rhine Valley. In addition, in one of our cases, nocturnal cooling introduces a smooth virtual topography that inhibits the formation of pronounced gravity waves and impedes the descent of foehn air parcels into the valley atmosphere. In summary, this study approaches a long-standing topic in foehn research from a new angle. Given the limitations of our model simulations, we did not attempt to unequivocally resolve the causes for the descent. Nevertheless, using online trajectories, we explicitly identified and characterized the descent of foehn. The innovative Lagrangian method enabled us to diagnose descent within a comprehensive dataset, encompassing multiple case studies and a wide range of different foehn regions. The findings highlight the benefits offered by the Lagrangian perspective, which not only complements but also substantially extends the previously predominant Eulerian perspective on the descent of foehn. [ABSTRACT FROM AUTHOR]

Natural modalities are often analysed from an abstract point of view where they are associated with putative laws of nature. However, the way possibilities are represented in physics is more complex. Lagrangian mechanics, for instance, involves two different layers of modalities: kinematical and dynamical possibilities. This paper examines the status of these two layers, both in the classical and quantum case. The quantum case is particularly problematic: we identify four possible interpretive options. The upshot is that a close inspection of the way possibilities are represented in physics could lead to new ways of thinking about natural modalities. [ABSTRACT FROM AUTHOR]

Accurate modeling of underwater vehicle dynamics is an essential component of various solutions designed to address a range of challenges involved in both the vehicle’s design and operation. Such models are usually parametric, including dynamic equations that simulate the vehicle’s response to various controls and environment conditions. They can be used to determine the vehicle’s capabilities, estimate the vehicle’s state in the absence of external communications, or to derive control signals to produce desired state responses. While a range of explicitly derived models have been commonly used in various applications, modeling the complex nonlinear dynamics using machine learning has recently attracted considerable interest. This topical review focuses on the integration of machine learning in underwater vehicle modeling, and covers two categories: artificial neural networks and non-parametric regression models. The first category includes recurrent neural networks and physics-informed neural network. They are trained to estimate model parameters, forces and moments from damping and disturbances, or to completely replace the dynamic model by outputting the expected state responses. The second category of the reviewed models covers support vector machines and Gaussian process models. These are non-parametric dynamic models and their training requirements are generally lower than ANN-based models. An overview of the theory behind each model is presented, along with examples of specific applications. The capabilities of each machine learning method are compared, and the challenges of their implementation for underwater vehicle dynamic modeling are discussed.

The relation between de Broglie's double-solution approach to quantum dynamics and the hydrodynamic pilot-wave system has motivated a number of recent revisitations and extensions of de Broglie's theory. Building upon these recent developments, we here introduce a rich family of pilot-wave systems, with a view to reformulating and studying de Broglie's double-solution program in the modern language of classical field theory. Notably, the entire family is local and Lorentz-invariant, follows from a variational principle, and exhibits time-invariant, two-way coupling between particle and pilot-wave field. We first introduce a variational framework for generic pilot-wave systems, including a derivation of particle-wave exchange of Noether currents. We then focus on a particular limit of our system, in which the particle is propelled by the local gradient of its pilot wave. In this case, we see that the Compton-scale oscillations proposed by de Broglie emerge naturally in the form of particle vibrations, and that the vibration modes dynamically adjust to match the Compton frequency in the rest frame of the particle. The underlying field dynamically changes its radiation patterns in order to satisfy the de Broglie relation p = ℏ k at the particle's position, even as the particle momentum p changes. The wave form and frequency thus evolve so as to conform to de Broglie's harmony of phases, even for unsteady particle motion. We show that the particle is always dressed with a Compton-scale Yukawa wavepacket, independent of its trajectory, and that the associated energy imparts a constant increase to the particle's inertial mass. Finally, we see that the particle's wave-induced Compton-scale oscillation gives rise to a classical version of the Heisenberg uncertainty principle. [ABSTRACT FROM AUTHOR]

The development of atmospheric blocks over the North Atlantic–European region can lead to extreme weather events like heat waves or cold air outbreaks. Despite their potential severe impact on surface weather, the correct prediction of blocking lifecycles remains a key challenge in current numerical weather prediction (NWP) models. Increasing evidence suggests that latent heat release in cyclones, the advection of cold air (cold air outbreaks, CAOs) from the Arctic over the North Atlantic, and associated air–sea interactions over the Gulf Stream are key processes contributing to the onset, maintenance, and persistence of such flow regimes. To better understand the mechanism connecting air–sea interactions over the Gulf Stream with changes in the large-scale flow, we focus on an episode between 20 and 27 February 2019, when a quasi-stationary upper-level ridge was established over western Europe accompanied by an intensified storm track in the northwestern North Atlantic. During that time, a record-breaking winter warm spell occurred over western Europe bringing temperatures above 20 ∘ C to the United Kingdom, the Netherlands, and northern France. The event was preceded and accompanied by the development of several rapidly intensifying cyclones that originated in the Gulf Stream region and traversed the North Atlantic. To explore the mechanistic linkage between the formation of this block and air–sea interactions over the Gulf Stream, we adopt a Lagrangian perspective, using kinematic trajectories. This allows us to study the pathways and transformations of air masses that form the upper-level potential vorticity anomaly and interact with the ocean front. We establish that more than one-fifth of these air masses interact with the Gulf Stream in the lower troposphere, experiencing intense heating and moistening over the region due to the frequent occurrence of CAOs behind the cold front of the cyclones. Trajectories moistened by the advection of cold air over a warm ocean by one cyclone later ascend into the upper troposphere with the ascending airstream of a subsequent cyclone, fueled by the strong surface fluxes. These findings highlight the importance of CAOs in the Gulf Stream region, indicating that their intense coupling between the ocean and atmosphere plays a role in block development. Additionally, they provide a mechanistic pathway linking air–sea interactions in the lower troposphere and the upper-level flow. [ABSTRACT FROM AUTHOR]

Future changes in extratropical cyclone structure and dynamics may lead to important impacts but are not yet fully understood. In the first part of this study, we have applied a composite approach together with potential vorticity (PV) inversion to study such changes in the dynamics of North Atlantic cyclones. Here, this is complemented with the help of a Lagrangian perspective, making use of air parcel trajectories to investigate the causes of altered PV anomalies as well as the role that cyclone airstreams play in shaping these changes. Intense cyclones in the extended winter seasons of two periods, 1990–2000 and 2091–2100, are studied in Community Earth System Model Large Ensemble (CESM-LE) simulations, and backward trajectories are calculated from the cyclone area as a basis to construct cyclone-centered composites of Lagrangian tendencies and their projected future changes. Our results show that diabatic processes on a timescale of 24 h shape the cyclones' low-level PV distribution and corroborate that the increasing moisture content along with enhanced ascent in warm conveyor belts leads to amplified latent heat release and larger low- and mid-level PV anomalies near the cyclone center in a warmer climate. In contrast, projected upper-level PV changes are due to a combination of several processes. These processes include cloud diabatic PV changes, anomalous PV advection, and likely also radiative PV generation in the lower stratosphere above the cyclone center. For instance, enhanced poleward advection is the primary reason for a projected decrease in upper-level PV anomalies south of the cyclone center. Warm conveyor belt outflow regions are projected to shift upward, but there is not robust change in the associated upper-level PV anomalies due to compensation between enhanced low-level PV generation and upper-level PV destruction. In summary, our two-part study points to future changes in the relative importance of different processes for the dynamics of intense North Atlantic cyclones in a warming climate, with important consequences for the near-surface wind pattern. In particular, a larger role of cloud diabatic processes is projected, affecting the cyclones through PV production in the lower troposphere. The role of other mechanisms, in particular radiative changes near the tropopause, should be investigated in more detail in future studies. [ABSTRACT FROM AUTHOR]

The extended guiding-centre Lagrangian equations of motion are derived by the Lie-transform perturbation method under the assumption of time-dependent and inhomogeneous electric and magnetic fields that satisfy the standard guiding-centre space–time orderings. Polarization effects are introduced into the Lagrangian dynamics by the inclusion of the polarization drift velocity in the guiding-centre velocity and the appearance of finite-Larmor-radius corrections in the guiding-centre Hamiltonian and guiding-centre Poisson bracket. [ABSTRACT FROM AUTHOR]

This paper addresses the robust stabilization problem of a cart–pole system. The controlled dynamics of this interconnected system are deduced by following the analytic framework of Lagrangian mechanics, and the residual terms are formulated as a bias depending on the angle and angular velocity. A geometric definition of Proportional–Integral–Derivative (PID) control algorithm is proposed, and a Lyapunov function is explicitly constructed through two stages of variable change. Local exponential stability of the stable equilibrium is proved, and a criterion for parameter tuning is provided by ensuring an exponential decrease in the Lyapunov function. Enlarging the control parameters to infinity allows for the extension of attraction region almost to the half circle. The effectiveness of geometric PID controller and the local exponential stability of the resulting close system are verified by simulating a numerical example. [ABSTRACT FROM AUTHOR]

By developing the previously proposed method of combining continuum mechanics with Einstein's field equations, it has been shown that the classic relativistic description, curvilinear description, and quantum description of the physical system may be reconciled using the proposed Alena Tensor. For a system with an electromagnetic field, the Lagrangian density equal to the invariant of the electromagnetic field was obtained, the vanishing four-divergence of canonical four-momentum appears to be the consequence of the Poynting theorem, and the explicit form of one of the electromagnetic four-potential gauges was introduced. The proposed method allows for further development with additional fields. [ABSTRACT FROM AUTHOR]

The inverse problem and universality theorems in Newtonian Mechanics have been extensively studied by R.M. Santilli, [11], [12]. In Analytical Mechanics or in the Higher-order Lagrangian Mechanics these problems were solved by M. Crampin [2], D. Krupka [5], M. de Leon [6] and many others. [8]. Generally, these studies are focused on the Euler-Lagrange equations. But, besides of the mentioned equations there exists also other important equations determined only by a Lagrangian of order k ≥ 1. For instance the integral curves of the canonical k-spray are given by a system of differential equations which depend on the Lagrangian L(x, y(1),..., y(k)) and has a geometrical meaning. Therefore in the higher-order Analytical Mechanics the inverse problem must be accordingly formulated. In this lecture we present the notion of k-osculator bundle, k-spray, h- paths and variational problem for the Lagrangians of order k, k ∈ N*. we formulate the inverse problem for each one of the systems E0i (L) =...= Ek-1i (L) = 0 and prove that the first one, E0i (L) = 0 is self-adjoint, while the others E¹i; (L) =...= Ek-1i (L) = 0 are non self-adjoint. Introducing the energies of order k and k-1 we discuss the laws of conservation and the Hamilton-Jacobi equations. [ABSTRACT FROM AUTHOR]

In this investigation, a magnetically coupled double-spring design is presented for harvesting low-level non-stationary random vibrational energy. The sensor relies on multimodal coupling between the translation and rotation of a two-spring magnet and coil system to widen the harvesting bandwidth. Energy methods are used to develop a model to characterize the electromechanical response of the system, the solution of which is obtained using stochastic techniques based on a particle swarm algorithm. This approach provides an efficient method to estimate system parameters that otherwise are difficult or impossible to determine with independent measurements. The experimental results demonstrate agreement with the theoretical predictions over a limited bandwidth. The sensor can effectively harvest non-stationary vibration energy down to 10−4 g within a limited bandwidth of 130–150 Hz. The sensor prototype has an operational volume of 2.6 cm3 with a calculated power density of 0.2 W/cm3. The sensor’s small size results in a coupling efficiency of approximately 6% across the tested bandwidth.

By developing the previously proposed method of combining continuum mechanics with Einstein’s field equations, it has been shown that the classic relativistic description, curvilinear description, and quantum description of the physical system may be reconciled using the proposed Alena Tensor. For a system with an electromagnetic field, the Lagrangian density equal to the invariant of the electromagnetic field was obtained, the vanishing four-divergence of canonical four-momentum appears to be the consequence of the Poynting theorem, and the explicit form of one of the electromagnetic four-potential gauges was introduced. The proposed method allows for further development with additional fields.

A geometric approach to the Herglotz problem is developed, based on the bundle of affine scalars on the configuration manifold of the given system. The environment, originally introduced to formalize the gauge structure of Lagrangian Mechanics [E. Massa, E. Pagani, and P. Lorenzoni, Transp. Theory Stat. Phys. 29, 69 (2000)], provides the natural setting for the representation of the Herglotz functional as well as for the study of its extremals. Various aspects of the problem are considered: the Lagrangian approach, leading to a generalization of the Poincaré-Cartan algorithm; the parametric approach, involving the introduction of an appropriate super-Lagrangian; the corresponding Hamiltonian and super-Hamiltonian counterparts; the relationship between the Herglotz problem and a constrained variational problem; the evaluation of the abnormality index [Massa et al., Int. J. Geom. Methods Mod. Phys. 12, 1550061 (2015)] of the resulting extremals; the gauge structure of the theory and the consequent existence of Herglotz's functionals gauge-equivalent to ordinary action functionals. [ABSTRACT FROM AUTHOR]

Robotics is a crucial technology of Industry 4.0 that offers a diverse array of applications in the industrial sector. However, the quality of a robot's manipulator is contingent on its stability, which is a function of the manipulator's parameters. In previous studies, stability has been evaluated based on a small number of manipulator parameters; as a result, there is not much information about the integration/optimal arrangement/combination of manipulator parameters toward stability. Through Lagrangian mechanics and the consideration of multiple parameters, a mathematical model of a modern manipulator is developed in this study. In this mathematical model, motor acceleration, moment of inertia, and deflection are considered in order to assess the level of stability of the ABB Robot manipulator of six degrees of freedom. A novel mathematical approach to stability is developed in which stability is correlated with motor acceleration, moment of inertia, and deflection. In addition to this, fuzzy logic inference principles are employed to determine the status of stability. The numerical data of different manipulator parameters are verified using mathematical approaches. Results indicated that as motor acceleration increases, stability increases, while stability decreases as moment of inertia and deflection increase. It is anticipated that the implementation of these findings will increase industrial output. [ABSTRACT FROM AUTHOR]

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Purpose: This article focuses on the low-velocity impact (LVI) output of carbon nanotubes (CNTs)' reinforcement circular plates, considering agglomeration size effect and clumping of CNTs' inner side of the agglomerations. Design/methodology/approach: A representative volume element (RVE) is used to determine the nanocomposite properties reinforced with agglomerated CNTs with random orientation. First-order shear deformation theory (FSDT) is used to obtain the motion equations of LVI analysis. These equations are handled by developing a Ritz method and Lagrangian mechanics. To extract the mass and stiffness matrices, terms with second and higher degrees are ignored. Findings: Formulation validation is performed by providing various examples, including comparisons with other research and ABAQUS FE code. The effects of agglomeration size, clumping of CNTs' inner side of the agglomerations, CNT volume fraction and impact location on the responses of impact load, projectile displacement and plate deflection are analytically studied. These achievements illuminate how the influence of agglomeration size is very small on the impact response. Also, the influence of clumping of CNTs' inner side of the agglomerations is significant, and as it increases, the displacement values and impact time increase, and the impact force decreases. Originality/value: In this article, to avoid additional calculations, the parameters of the mass matrix and the stiffness coefficients are linearized to obtain the equations of motion of the impact on the circular plate. [ABSTRACT FROM AUTHOR]

Some simple models of a child swinging on a playground swing are presented. These are analyzed using techniques from Lagrangian mechanics with a twist: the child changes the configuration of the system by sudden movements of their body at key moments in the oscillation. This can lead to jumps in the generalized coordinates describing the system and/or their velocities. Jump conditions can be determined by integrating the Euler--Lagrange equations over a short time interval and then taking the limit as this time interval goes to zero. These models give insights into strategies used by swingers, and answer such vexed questions such as whether it is possible for the swing to go through a full 360 ° turn over its pivot. A model of an instability at the pivot observed by Colin Furze in a rigid swing constructed to rotate through 360 ° is also described. This uses a novel double pendulum configuration in which the two components of the pendulums are constrained to move in orthogonal planes. [ABSTRACT FROM AUTHOR]

Originally developed for solving fluid dynamics problems, a Lagrangian particle method called the consistent particle method (CPM) is expanded to solid mechanics problems in this article. The motions of particles are governed by the mass conservation and dynamics equations which are solved by the second‐order predictor–corrector time stepping scheme. The spatial derivatives of variables required in solving the governing equations and computing strain increments are obtained by Taylor series expansion. Stress components are updated from strain components according to the material constitutive relation. The physical density of material is directly computed without using particle number density. No additional particles are needed for the simulation of boundaries. For boundary surface that is partially loaded, the inverse distance weighting method is applied to deal with the stress singularity problem due to sudden change of stresses. Numerical examples including shock wave propagation and large deformation of footing penetration are presented to demonstrate the capabilities of CPM in solving solid mechanics problems. [ABSTRACT FROM AUTHOR]

Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton–Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by L. Schwartz and P.-A. Meyer, to construct second-order counterparts of those classical structures. These will allow us to study symmetries of stochastic differential equations (SDEs), to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton–Jacobi–Bellman (HJB) equations. Indeed, stochastic prolongation formulae will be derived to study symmetries of SDEs and mixed-order Cartan symmetries. Stochastic Hamilton’s equations will follow from a second-order symplectic structure and canonical transformations will lead to the HJB equation. A stochastic variational problem on Riemannian manifolds will provide a stochastic Euler–Lagrange equation compatible with HJB one and equivalent to the Riemannian version of stochastic Hamilton’s equations. A stochastic Noether’s theorem will also follow. The inspirational example, along the paper, will be the rich dynamical structure of Schrödinger’s problem in optimal transport, where the latter is also regarded as a Euclidean version of hydrodynamical interpretation of quantum mechanics. [ABSTRACT FROM AUTHOR]

We give a geometric interpretation for the principle of stationary action in classical Lagrangian particle mechanics. In a nutshell, the difference of the action along a path and its variation effectively "counts" the possible evolutions that "go through" the area enclosed. If the path corresponds to a possible evolution, all neighbouring evolutions will be parallel, making them tangent to the area enclosed by the path and its variation, thus yielding a stationary action. This treatment gives a full physical account of the geometry of both Hamiltonian and Lagrangian mechanics which is founded on three assumptions: determinism and reversible evolution, independence of the degrees of freedom and equivalence between kinematics and dynamics. The logical equivalence between the three assumptions and the principle of stationary action leads to a much cleaner conceptual understanding. [ABSTRACT FROM AUTHOR]

In this article, the problems of the fractional calculus of variations are discussed based on generalized fractional operators, and the corresponding Lagrange equations are established. Then, the Noether symmetry method and the perturbation to Noether symmetry are analyzed in order to find the integrals of the equations. As a result, the conserved quantities and the adiabatic invariants are obtained. Due to the universality of the generalized fractional operators, the results achieved here can be used to solve other specific problems. Several examples are given to illustrate the universality of the methods and results. [ABSTRACT FROM AUTHOR]

The Eringen's nonlocal elastica equation does not possess a Lagrangian formulation. In this article, we find a variational integrating factor which enables us to provide a Lagrangian and Hamiltonian structure associated to this equation. Explicit expressions of the solutions in terms of elliptic integrals of the first kind are then deduced. We then derive discrete version of the Eringen's nonlocal elastica preserving the Lagrangian and Hamiltonian structure and compare it with Challamel's and co-worker definition of a discrete Eringen's nonlocal elastica. [ABSTRACT FROM AUTHOR]

In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete ℓ 2 -norm error O (h 2 + △ x 4 + △ x p + 1 / h) by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme. In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete -norm error by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme. [ABSTRACT FROM AUTHOR]

We show that the solution of Cauchy problem for the classical ODE m y ″ = f can be obtained as the limit of minimizers of exponentially weighted convex variational integrals. This complements the known results about weighted inertia-energy approach to Lagrangian mechanics and hyperbolic equations. [ABSTRACT FROM AUTHOR]

The Bayesian approach of cognitive science largely takes the position that evolution drives perception to produce precepts that are veridical. However, some efforts utilizing evolutionary game theory simulations have shown that perception is more likely based on a fitness function, which promotes survival rather than promoting perceptual truth about the environment. Although these findings do not correspond well with the standard Bayesian approach to cognition, they may correspond with a behavioral functional contextual approach that is ontologically neutral (a-ontological). This approach, formalized through a post-Skinnerian account of behaviorism called relational frame theory (RFT), can, in fact, be shown to correspond well with an evolutionary fitness function, whereby contextual functions form that corresponds to a fitness function interface of the world. This fitness interface approach therefore may help provide a mathematical description for a functional contextual interface of phenomenological experience. Furthermore, this more broadly fits with a neurological active inference approach based on the free-energy principle (FEP) and more broadly with Lagrangian mechanics. These assumptions of how fitness beats truth (FBT) and FEP correspond to RFT are then discussed within a broader multidimensional and evolutionary framework called the extended evolutionary meta-model (EEMM) that has emerged out of the functional contextual behavioral science literature to incorporate principles of cognition, neurobiology, behaviorism, and evolution and are discussed in the context of a novel RFT framework called “Neurobiological and Natural Selection Relational Frame Theory” (N-frame). This framework mathematically connects RFT to FBT, FEP, and EEMM within a single framework that expands into dynamic graph networking. This is then discussed for its implications of empirical work at the non-ergodic process-based idiographic level as applied to individual and societal level dynamic modeling and clinical work. This discussion is framed within the context of individuals that are described as evolutionary adaptive and conscious (observer-self) agents that minimize entropy and can promote a prosocial society through group-level values and psychological flexibility. [ABSTRACT FROM AUTHOR]

Motivated by the concept of "location uncertainty", initially introduced in , a scheme is sought to perturb the "location" of a state variable at every forecast time step. Further considering Brenier's theorem , asserting that the difference of two positive density fields on the same domain can be represented by a transportation map, we demonstrate that the perturbations consistently define a stochastic partial differential equation (SPDE) from the original PDE. It ensues that certain quantities, up to the user, are conserved at every time step. Remarkably, derivations following both the SALT (stochastic advection by Lie transport;) and LU (location uncertainty;) settings can be recovered from this perturbation scheme. Still, it offers broader applicability since it does not explicitly rely on Lagrangian mechanics or Newton's laws of force. For illustration, a stochastic version of the thermal shallow water equation is presented. [ABSTRACT FROM AUTHOR]

Underwater explosions that threaten the hull integrity and proper operation of navy ships' systems and subsystems are considered from the viewpoint of damage estimation. The commercial code LS-DYNA with the Arbitrary Lagrangian-Eulerian numerical technique is used to analyze the shock effects on structural components of a ship-like form for three different scenarios. Computed maximum permanent deformations are matched with the corresponding keel shock factor (KSF) values for searching a functional relationship of a linear form. Although observed for limited simulations and therefore should be viewed with caution, the linear dependency of the maximum permanent deformations on the KSF values implies that the KSF may be a simple indicator for estimating the extent of damage due to underwater blasts. [ABSTRACT FROM AUTHOR]

Motivated by the concept of "location uncertainty", initially introduced in Mémin (2014), a scheme is sought to perturb the "location" of a state variable at every forecast time step. Further considering Brenier's theorem Brenier (1991), asserting that the difference of two positive density fields on the same domain can be represented by a transportation map, perturbations are demonstrated to consistently define a SPDE from the original PDE. It ensues that certain quantities, up to the user, are conserved at every 5 time step. Remarkably, derivations following both the SALT Holm (2015) and LU Mémin (2014); Resseguier et al. (2016) settings, can be recovered from this perturbation scheme. Still, it opens broader applicability since it does not explicitly rely on Lagrangian mechanics or Newton's laws of force. For illustration, a stochastic version of the thermal shallow water equation is presented. [ABSTRACT FROM AUTHOR]

Deep learning has been widely used within learning algorithms for robotics. One disadvantage of deep networks is that these networks are black-box representations. Therefore, the learned approximations ignore the existing knowledge of physics or robotics. Especially for learning dynamics models, these black-box models are not desirable as the underlying principles are well understood and the standard deep networks can learn dynamics that violate these principles. To learn dynamics models with deep networks that guarantee physically plausible dynamics, we introduce physics-inspired deep networks that combine first principles from physics with deep learning. We incorporate Lagrangian mechanics within the model learning such that all approximated models adhere to the laws of physics and conserve energy. Deep Lagrangian Networks (DeLaN) parametrize the system energy using two networks. The parameters are obtained by minimizing the squared residual of the Euler–Lagrange differential equation. Therefore, the resulting model does not require specific knowledge of the individual system, is interpretable, and can be used as a forward, inverse, and energy model. Previously these properties were only obtained when using system identification techniques that require knowledge of the kinematic structure. We apply DeLaN to learning dynamics models and apply these models to control simulated and physical rigid body systems. The results show that the proposed approach obtains dynamics models that can be applied to physical systems for real-time control. Compared to standard deep networks, the physics-inspired models learn better models and capture the underlying structure of the dynamics. [ABSTRACT FROM AUTHOR]

Projection-based embedding (PbE) is an exact embedding method within density-functional theory (DFT) that has received increasing attention in recent years. Several different variants have been described in the literature, but no systematic comparison has been presented so far. The truncation of the basis is critical for the efficiency of this class of approaches. Here, we employ a basis-set truncation scheme previously used for level-shift embedding in a top-down fashion, and we present an own basis-set extension scheme for bottom-up type PbE. We compare its accuracy for the level-shift technique [Manby et al., J. Chem. Theory Comput. 8, 2564–2568 (2012)] and an empirically corrected variant, the external-orthogonality approach by Khait and Hoffmann [Annu. Rep. Comput. Chem. 8, 53–70 (2012)] and the approach based on the Huzinaga equation transferred to the DFT context [Hégely et al., J. Chem. Phys. 145, 064107 (2016)]. Concerning the reproduction in total energies, we show that the Huzinaga method yields the most stable results concerning a basis-set truncation in top-down embedding. For the practically more relevant calculation of energy differences, the efficient level-shift technique yields very promising results due to error cancellation. In bottom-up embedding, we observe convergence issues in cases where constraints in the Lagrange formalism cannot be fulfilled due to basis-set incompleteness. [ABSTRACT FROM AUTHOR]

For the ⟨ 1 1 ¯ 0 ⟩ screw dislocation (nondissociated) in silicon, there are three types of core structures that are, respectively, referred to as the shuffle-set dislocation (A-core), glide-set dislocation (C-core), and mix-set dislocation (B-core). Each type of core further displays a planar or nonplanar (fourfold) configuration. In this paper, in the context of the generalized Peierls theory, the energy functional and equation of equilibrium of a screw dislocation with a nonorthogonal fourfold structure is derived and is used to investigate the structure and energy of the ⟨ 1 1 ¯ 0 ⟩ screw dislocation in silicon. We find that the energy of the shuffle-set dislocation with a fourfold core is considerably lower than that of the other types of cores. Thus, the shuffle-set dislocation with the fourfold core is the most stable. The glide-set dislocation with a planar core has the highest energy and is the most unstable. As the most stable structure, the shuffle-set screw dislocation with the fourfold core was responsible for wavy slip traces observed at low temperature. [ABSTRACT FROM AUTHOR]

Studying worked examples has been shown by extensive research to be an effective method for learning to solve well-structured problems in physics and mathematics. The effectiveness of learning with worked examples has been demonstrated and documented in many research projects. In this work, we propose a new four-step approach for teaching with worked examples that includes writing explanations and finding and correcting errors. This teaching method can even be implemented in courses in which homework performance constitutes part of the grading system. This four-step approach is illustrated in the context of Lagrangian mechanics, which is ideal for the application of worked examples due to its universal approach to solve problems. [ABSTRACT FROM AUTHOR]

Unfortunately, the Hamiltonian mechanics of degenerate Lagrangian systems is usually presented as a mere recipe of Dirac, with no explanation as to how it works. It then comes to discussing conjectures of whether all primary constraints correspond to gauge symmetries, and it goes all the way to absolutely wrong claims such as the statement that electrodynamics or gravity only have two physical components each, with others being spurious. One has to be very careful because non-dynamical, or constrained, does not mean unphysical. In this article, I give a pedagogical introduction to the degenerate Hamiltonian systems, showing both very simple mechanical examples and general arguments about how it works. For the familiar field theory models, I explain why the gauge freedom there "hits twice" in the sense of producing twice as many first-class constraints as gauge symmetries, and why primary, and only primary, constraints should be put into the total Hamiltonian. [ABSTRACT FROM AUTHOR]

In this work, a dynamic model is proposed to simulate the drilling and steering process of an autonomous burrowing mole to access scientific samples from the deep subsurface of the Moon. The locomotive module is idealized as a rigid rod. The characteristic parameters are considered, including the length, cross-section diameter, and centroid of a cylindrical rod. Based on classical Lagrangian mechanics, a 3-DOF dynamic model for the locomotion of this autonomous device is developed. By introducing resistive force theory, the interaction scheme between the locomotive body and the lunar regolith is described. The effects of characteristic parameters on resistive forces and torques are studied and discussed. Proportional-derivative control strategies are introduced to calculate the tracking control forces following a planned trajectory. The simulation results show that this method provides a reliable manipulation of a mole-type robot to avoid obstacles during the tracking control process in layered sediments. Overall, the proposed reduced-order model is able to simulate the operating and controlling scenarios of an autonomous burrowing robot in lunar subsurface environments. This model provides intuitive inputs to plan the space missions of a drilling robot to extract subsurface samples on an extraterrestrial planet such as the Moon or Mars. [ABSTRACT FROM AUTHOR]

Mediterranean cyclones are extratropical cyclones, typically of smaller size and weaker intensity than other cyclones that develop over the main open ocean storm tracks. Nevertheless, Mediterranean cyclones can attain high intensities, even comparable to the ones of tropical cyclones, and thus cause large socioeconomic impacts in the densely populated coasts of the region. After cyclogenesis takes place, a large variety of processes are involved in the cyclone's development, contributing with positive and negative potential vorticity (PV) changes to the lower-tropospheric PV anomalies in the cyclone center. Although the diabatic processes that produce these PV anomalies in Mediterranean cyclones are known, it is still an open question whether they occur locally within the cyclone itself or remotely in the environment (e.g., near high orography) with a subsequent transport of high-PV air into the cyclone center. This study introduces a Lagrangian method to determine the origin of the lower-tropospheric PV anomaly, which is applied climatologically to ERA5 reanalysis and to 12 monthly simulations, performed with the integrated forecasting system (IFS) model. We define and quantify so-called "cyclonic" and "environmental" PV and find that the main part of the lower-tropospheric PV anomaly (60 %) is produced within the cyclone, shortly prior (-12 h) to the cyclones' mature stage. Nevertheless, in 19.5 % of the cyclones the environmental PV production near the mountains surrounding the Mediterranean Basin plays a significant role in forming the low-tropospheric PV anomaly and therefore in determining the intensity of these cyclones. The analysis of PV tendencies from the IFS simulations reveals that the major PV production inside the cyclone is typically due to convection and microphysics, whereas convection and turbulent momentum tendencies cause most of the positive PV changes in the environment. [ABSTRACT FROM AUTHOR]

We construct a formal gradient flow structure for phase-field evolution coupled to mechanics in Lagrangian coordinates, present common ways to couple the evolution, and provide an incremental minimization strategy. While the usual presentation of continuum mechanics is intentionally brief, we construct an extensible functional analytical framework and a discretization approach that preserves the underlying variational structure. We consider phase separation and swelling of gels and then study stationary states of multiphase systems with surface tension and contact lines and show the robustness of the general approach for large deformations. We highlight differences between compressible and incompressible models and discuss issues of the sharp-interface limit for different magnitudes of the Cahn--Hilliard mobility. [ABSTRACT FROM AUTHOR]